Topological library. Part 3: Spectral sequences in topology. Transl. by V. P. Golubyatnikov.

*(English)*Zbl 1264.55002“Topological Library” is a three-volume edition of classical papers in algebraic and differential topology published in the period 1950 to 1970. Apart from the fact that the papers collected together in these volumes have been of fundamental importance for the development of topology at that time, and therefore have been of pioneering, truly epoch-making impact back then, there has never been an appropriate exposition of their results in the relevant textbook literature up to now. In view of this particular situation, the editors of this three-volume series have done a highly valuable and very rewarding service to the mathematical community by arranging a large number of such significant articles in book form, thereby having provided translations into English of several of the original papers for the first time at all.

While the first volume of the series presented eight classical articles emphasizing cobordism theory and its applications [Topological library. Part 1: Cobordisms and their applications. Series on Knots and Everything 39. Hackensack, NJ: World Scientific. xiv, 369 p. (2007; Zbl 1121.55003)], the second volume comprised six earlier papers on homotopy theory and characteristic classes of smooth manifolds [Topological library. Part 2: Characteristic classes and smooth structures on manifolds. Series on Knot and Everything 44. Hackensack, NJ: World Scientific. xiv, 261 p. (2010; Zbl 1195.57004)].

The book under review is the third (and final) volume of the edition “Topological Library”. This collection contains nine classical papers on topics in algebraic and differential topology published between 1951 and 1967, with a focus on spectral sequences in topology, (co)homology theory of special topological spaces and manifolds, and the structure of the Steenrod algebra. More precisely, the following important articles are here reprinted, partially translated into English, and supplied with commenting footnotes by the editors:

(1) J.-P. Serre, Ann. Math. (2) 54, 425–505 (1951; Zbl 0045.26003); (2) J.-P. Serre, Ann. Math. (2) 58, 258–294 (1953; Zbl 0052.19303); (3) J.-P. Serre, Comment. Math. Helv. 27, 198–232 (1953; Zbl 0052.19501); (4) A. Borel, Ann. Math. (2) 57, 115–207 (1953; Zbl 0052.40001); (5) A. Borel, Comment. Math. Helv. 27, 165–197 (1953; Zbl 0052.40301); (6) J. Milnor, Ann. Math. (2) 67, 150–171 (1958; Zbl 0080.38003); (7) J. F. Adams, Comment. Math. Helv. 32, 180–214 (1958; Zbl 0083.17802); (8) M. F. Atiyah and F. Hirzebruch, Proc. Sympos. Pure Math. 3, 7–38 (1961; Zbl 0108.17705); (9) S. P. Novikov, Izv. Akad. Nauk SSSR, Ser. Mat. 31, 855–951 (1967; Zbl 0169.54503).

No doubt, it is utmost useful to have these (interrelated) classics gathered together in one volume. This facilitates the study of the originals considerably, all the more as numerous editorial hints provide additional guidance. In this regard, the entire edition represents an invaluable source book for both students and researchers in the field.

While the first volume of the series presented eight classical articles emphasizing cobordism theory and its applications [Topological library. Part 1: Cobordisms and their applications. Series on Knots and Everything 39. Hackensack, NJ: World Scientific. xiv, 369 p. (2007; Zbl 1121.55003)], the second volume comprised six earlier papers on homotopy theory and characteristic classes of smooth manifolds [Topological library. Part 2: Characteristic classes and smooth structures on manifolds. Series on Knot and Everything 44. Hackensack, NJ: World Scientific. xiv, 261 p. (2010; Zbl 1195.57004)].

The book under review is the third (and final) volume of the edition “Topological Library”. This collection contains nine classical papers on topics in algebraic and differential topology published between 1951 and 1967, with a focus on spectral sequences in topology, (co)homology theory of special topological spaces and manifolds, and the structure of the Steenrod algebra. More precisely, the following important articles are here reprinted, partially translated into English, and supplied with commenting footnotes by the editors:

(1) J.-P. Serre, Ann. Math. (2) 54, 425–505 (1951; Zbl 0045.26003); (2) J.-P. Serre, Ann. Math. (2) 58, 258–294 (1953; Zbl 0052.19303); (3) J.-P. Serre, Comment. Math. Helv. 27, 198–232 (1953; Zbl 0052.19501); (4) A. Borel, Ann. Math. (2) 57, 115–207 (1953; Zbl 0052.40001); (5) A. Borel, Comment. Math. Helv. 27, 165–197 (1953; Zbl 0052.40301); (6) J. Milnor, Ann. Math. (2) 67, 150–171 (1958; Zbl 0080.38003); (7) J. F. Adams, Comment. Math. Helv. 32, 180–214 (1958; Zbl 0083.17802); (8) M. F. Atiyah and F. Hirzebruch, Proc. Sympos. Pure Math. 3, 7–38 (1961; Zbl 0108.17705); (9) S. P. Novikov, Izv. Akad. Nauk SSSR, Ser. Mat. 31, 855–951 (1967; Zbl 0169.54503).

No doubt, it is utmost useful to have these (interrelated) classics gathered together in one volume. This facilitates the study of the originals considerably, all the more as numerous editorial hints provide additional guidance. In this regard, the entire edition represents an invaluable source book for both students and researchers in the field.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

55-03 | History of algebraic topology |

01A75 | Collected or selected works; reprintings or translations of classics |

55Rxx | Fiber spaces and bundles in algebraic topology |

55Q05 | Homotopy groups, general; sets of homotopy classes |

55S10 | Steenrod algebra |

55N10 | Singular homology and cohomology theory |

55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |

57R22 | Topology of vector bundles and fiber bundles |

57T15 | Homology and cohomology of homogeneous spaces of Lie groups |